Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
integral inequality; bipartite graph; graph homomorphism; Sidorenko's conjecture
Summary:
We study lower estimates for integral fuctionals for which the structure of the integrand is defined by a graph, in particular, by a bipartite graph. Functionals of such kind appear in statistical mechanics and quantum chemistry in the context of Mayer's transformation and Mayer's cluster integrals. Integral functionals generated by graphs play an important role in the theory of graph limits. Specific kind of functionals generated by bipartite graphs are at the center of the famous and much studied Sidorenko's conjecture, where a certain lower bound is conjectured to hold for every bipartite graph. In the present paper we work with functionals more general and lower bounds significantly sharper than those in Sidorenko's conjecture. In his 1991 seminal paper, Sidorenko proved such sharper bounds for several classes of bipartite graphs. To obtain his result he used a certain way of ``gluing'' graphs. We prove his inequality for a new class of bipartite graphs by defining a different type of gluing.
References:
[1] Bernardi, O.: Solution to a combinatorial puzzle arising from Mayer's theory of cluster integrals. Sémin. Lothar. Comb. 59 (2007), Article No. B59e, 10 pages. MR 2465401 | Zbl 1193.05012
[2] Bogachev, V. I.: Measure Theory. Vol. I, II. Springer, Berlin (2007). DOI 10.1007/978-3-540-34514-5 | MR 2267655 | Zbl 1120.28001
[3] Conlon, D., Fox, J., Sudakov, B.: An approximate version of Sidorenko's conjecture. Geom. Funct. Anal. 20 (2010), 1354-1366. DOI 10.1007/s00039-010-0097-0 | MR 2738996 | Zbl 1228.05285
[4] Conlon, D., Kim, J. H., Lee, C., Lee, J.: Some advances on Sidorenko's conjecture. J. London Math. Soc. 98 (2018), 593-608. DOI 10.1112/jlms.12142 | MR 3893193 | Zbl 07000630
[5] Conlon, D., Lee, J.: Finite reflection groups and graph norms. Adv. Math. 315 (2017), 130-165. DOI 10.1016/j.aim.2017.05.009 | MR 3667583 | Zbl 1366.05107
[6] Hatami, H.: Graph norms and Sidorenko's conjecture. Isr. J. Math. 175 (2010), 125-150. DOI 10.1007/s11856-010-0005-1 | MR 2607540 | Zbl 1227.05183
[7] Kaouche, A., Labelle, G.: Mayer and Ree-Hoover weights, graph invariants and bipartite complete graphs. P.U.M.A., Pure Math. Appl. 24 (2013), 19-29. MR 3197094 | Zbl 1313.05019
[8] Kim, J. H., Lee, C., Lee, J.: Two approaches to Sidorenko's conjecture. Trans. Am. Math. Soc. 368 (2016), 5057-5074. DOI 10.1090/tran/6487 | MR 3456171 | Zbl 1331.05220
[9] Král', D., Martins, T. L., Pach, P. P., Wrochna, M.: The step Sidorenko property and non-norming edge-transitive graphs. Available at https://arxiv.org/abs/1802.05007 MR 3873870
[10] Labelle, G., Leroux, P., Ducharme, M. G.: Graph weights arising from Mayer's theory of cluster integrals. Sémin. Lothar. Comb. 54 (2005), Article No. B54m, 40 pages. MR 2341745 | Zbl 1188.82007
[11] Li, J. L., Szegedy, B.: On the logarithmic calculus and Sidorenko's conjecture. Available at https://arxiv.org/abs/1107.1153
[12] Lovász, L.: Large Networks and Graph Limits. Colloquium Publications 60. American Mathematical Society, Providence (2012). DOI 10.1090/coll/060 | MR 3012035 | Zbl 1292.05001
[13] Mayer, J. E., Mayer, M. Göppert: Statistical Mechanics. J. Wiley and Sons, New York (1940),\99999JFM99999 66.1175.01. MR 0674819
[14] Royden, H. L.: Real Analysis. Macmillan Publishing, New York (1988). MR 1013117 | Zbl 0704.26006
[15] Sidorenko, A. F.: Inequalities for functionals generated by bipartite graphs. Discrete Math. Appl. 2 (1991), Article No. 489-504 English. Russian original translation from Diskretn. Mat. 3 1991 50-65. DOI 10.1515/dma.1992.2.5.489 | MR 1138091 | Zbl 0787.05052
[16] Sidorenko, A.: A correlation inequality for bipartite graphs. Graphs Comb. 9 (1993), 201-204. DOI 10.1007/BF02988307 | MR 1225933 | Zbl 0777.05096
[17] Szegedy, B.: An information theoretic approach to Sidorenko's conjecture. Available at https://arxiv.org/abs/1406.6738
Partner of
EuDML logo